External Ellipsoidal Harmonics for the Dunkl–Laplacian⋆

Following the book [9] by Dunkl and Xu and the paper [23] by Xu we will assume that αj ≥ 0 for each j = 0, 1, . . . , k although one would expect that the range of validity can be extended analytically to the domain αj > −12 for each j. We will also exclude the case k = 1, α0 = α1 = 0 because we want the constant μ defined below in (1.5) to be positive. The parity vector p = (p0, p1, . . . , pk) has components in {0, 1} and indicates that Fn,p and Gn,p have parity p which means that they are sums of monomials x p0+2q0 0 x p1+2q1 1 · · · x pk+2qk k with qj ∈ N0 = {0, 1, 2, . . . }. The vector n = (n1, n2, . . . , np) counts the zeros of the corresponding Stieltjes quasi-polynomials inside k adjacent open intervals. If we set m = 2|n|+ |p| :=